An Asymptotic Analysis of Space Charge Layers in a Mathematical Model of a Solid Electrolyte
We review a model for a solid electrolyte derived under thermodynamics principles. We non-dimensionalise and scale the model to identify small parameters, where we identify a scaling that controls the width of the space-charge layer in the electrolyte. We present asymptotic analyses and numerical solutions for the one dimensional zero charge flux equilibrium problem. We introduce an auxiliary variable to remove singularities from the domain in order to facilitate robust numerical simulations. From the asymptotics we identify three distinct regions: the bulk, boundary layers, and intermediate layers. The boundary and intermediate layers form the space charge layer of the solid electrolyte, which we can further distinguish as strong and weak space-charge-layers respectively. The weak space-charge-layer is characterised by a length, $λ$, which is equivalent to the Debye length of a standard liquid electrolyte. The strong space-charge-layer is characterised by a scaled Debye length, which is larger than $λ$. We find that both layers exhibit distinct behaviour, we see quadratic behaviour in the strong space-charge-layer and exponential behaviour in the weak space-charge-layer. We find that matching between these two asymptotic regimes is not standard and we implement a pseudo-matching approach to facilitate the transition between the quadratic and exponential behaviours. We demonstrate excellent agreement between asymptotics and simulation.